Coherence of assistance is another quantifier which quantifies the coherence that can be extracted assisted by another party under local measurements and classical communication [1]}.
Suppose Alice holds a state \(\rho ^A=\sum _k p_k |\psi _k\rangle \langle \psi _k|\)  with coherence \(C(\rho ^A)\) . Bob holds another part of the purified state of \(\rho ^A\) . The joint state between Alice and Bob is \(\sum _k p_k |\psi _k\rangle _A \otimes |k\rangle _B\) . Bob performs local measurements \(\lbrace |k\rangle \langle k|\rbrace \)  and informs Alice the measurement outcomes by classical communication. Alice's quantum state will be in a pure state ensemble \(\lbrace  p_k,\ |\psi _k\rangle \langle \psi _k|\rbrace \)  with average coherence \(\sum _k p_k C(|\psi _k\rangle \langle \psi _k|)\) . The process is called assisted coherence distillation. The maximum average coherence is called the coherence of assistance which quantifies the one-way coherence
distillation rate [1]}. The coherence of assistance is always greater than or equal to the coherence measure. But it is still not clear whether one can always obtain more coherence with the help of another party. Our answer in this paper is that one can always obtain more coherence for the full ranked quantum states.
